Addition of rational expressions

Definition

The addition of rational expressions involves combining two or more fractions that have polynomial numerators and denominators. To successfully add these expressions, one must find a common denominator, which allows the numerators to be combined. This process is essential in simplifying complex algebraic problems and is foundational in manipulating rational equations.

5 Must Know Facts For Your Next Test

1. To add rational expressions, first identify the least common denominator (LCD) for the denominators involved.
2. When the denominators are already the same, simply add the numerators while keeping the common denominator.
3. If the denominators differ, convert each expression to have the LCD before combining them.
4. After combining, simplify the resulting expression by factoring and reducing any common terms in the numerator and denominator.
5. It's important to check for restrictions on the variable values that would make any denominator equal to zero, as these values are excluded from the domain.

Review Questions

• How do you determine the least common denominator when adding two rational expressions?
  • To determine the least common denominator (LCD) when adding two rational expressions, first factor each denominator completely into its prime factors. Then, for each unique factor present in any denominator, take the highest power of that factor. The product of these factors will give you the LCD, which you can use to rewrite each rational expression before adding them together.
• What steps should be taken after finding a common denominator to add two rational expressions?
  • After finding a common denominator, rewrite each rational expression so that they both have this shared denominator. Then, combine the numerators by adding them together while keeping the common denominator intact. Finally, simplify the resulting expression by factoring if possible and reducing any common factors between the numerator and denominator.
• Evaluate how adding rational expressions can affect their overall simplification in solving equations.
  • Adding rational expressions can significantly impact overall simplification when solving equations because it allows you to combine like terms and manage complex algebraic fractions efficiently. By finding a common denominator and combining these expressions, you can often simplify an equation into a more manageable form. This can lead to easier solutions and clearer insight into the relationships between variables within equations. However, care must be taken to simplify fully, as this affects subsequent calculations and solutions.

Related terms

Common Denominator: A common denominator is a shared multiple of the denominators of two or more fractions, which allows for their addition or subtraction.

Rational Expression: A rational expression is a fraction in which both the numerator and denominator are polynomials.

Polynomial: A polynomial is an algebraic expression that consists of variables raised to whole number exponents and their coefficients, combined using addition, subtraction, and multiplication.